ALMA Memo 523

Simulation of Atmospheric Phase Correction

Combined With Instrumental Phase Calibration

Using Fast Switching

Originally LAMA Memo 803 2004 Jan 239

M. A. Holdaway and L. D’AddarioNational Radio Astronomy Observatory

949 N. Cherry Ave, Tucson AZ 85721-0655contact author: mholdawa@nrao.edu

LAMA Memo 803Simulation of Atmospheric Phase Correction

Combined With Instrumental Phase CalibrationUsing Fast Switching

M.A. Holdaway and L. D’AddarioNational Radio Astronomy Observatory

949 N. Cherry Ave.Tucson, AZ 85721-0655

email: mholdawa@nrao.edu

January 23, 2004

Abstract

We present very detailed simulations of fast switching phase correction which treats, asaccurately as we can, fast switching at all frequency bands available to ALMA, with thecalibration being performed at either the target frequency or at 90 GHz.

This work leads us to the following results and potential problems:

• Fast switching will typically have efficiencies (considering both residual decorrelationand time spent calibrating) ranging from 0.70 at the highest frequencies to 0.90 atthe lowest.

• Residual phase errors will range from 35 deg rms at the highest frequencies to 15 degrms at the lowest frequencies.

• For target frequencies above 345 GHz, calibrating at 90 GHz (where the sources arebrighter and the sensitivity is better, but requires an extra calibration observationto determine the difference in the instrumental phases at the two frequencies) willresult in about a 5% increase in efficiency over calibrating at the target frequency.It appears that below 345 GHz, there is little gain in performing the calibration at90 GHz, and we may very well calibrate at the target frequency.

• Performing fast switching at 90 GHz, rather than at the target frequency, only im-proves the efficiency by about 5% in the sub-millimeter, and may be deemed too muchtrouble for too little gain.

• Problems with modeling dispersive delay in the sub-millimeter regime, or dry path-length fluctuations together with even well-modeled dispersive delay, could push usto perform calibrator observations at the target frequency above 345 GHz.

• Calibrator sources (ie, flat spectrum quasars) might not look the same at 90 GHz andthe target frequencies because of dust emission at the higher frequency, or becausethe position of the core changes with frequency. These effects could make it difficultto calibrate at a frequency other than the target frequency.

• The fast switching pointing specification (being able to move 1.5 deg in 1.5 sec witha 3 arcsec residual pointing error) is not acceptable for mosaicing or sub-millimeter

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observations, and we need to study how quickly the antenna returns to an acceptablepointing residual.

• The specification on changing frequencies is 1.5 s, which was intended to match thefast switching antenna motion specification. If this could be shaved to 1 s, we couldprobably make good use of that improved frequency change time.

• At the best 5% conditions in the joint opacity/phase stability distributions, we de-termine that the residual phase jitter over short times (ie, 20 s or less) is 111 fs rmsdelay.

• Also at the best 5% atmospheric conditions, simulations indicate that longer timescale(ie, 300 s) atmosphereic delay variation will produce residual errors in visibility phasecorresponding to about 32 fsec rms.

1 Introduction

1.1 Phase Errors are Really Bad

Phase fluctuations due to the atmosphere are really bad. If unchecked, they will result in:

• decorrelation, which means a loss of sensitivity.

• the decorrelation will actually be variable, leading to visibility amplitude fluctuations.These in turn will complicate the flux calibration and limit the image quality.

• phase errors, which basically limit our ability to determine where anything really is,messing up our images and limiting our resolution.

Chajnantor may be one of the best millimeter and sub-millimeter sites in the world, but letme let you in on a little secret: for the median phase stability conditions measured by SimonRadford’s 300 m baseline interferometer, if we were to leave the phase fluctuations uncorrectedat 230 GHz on that same 300 m baseline, the coherence would only be 0.5! That’s morethan half of ALMA’s sensitivity down the drain for half of the time, at the relatively lowfrequency of 230 GHz and relatively short baselines of 300 m. Remember this: Chajnantor isan excellent site as judged by transparency, but when you look at the phase stability, ALMAis only marginally better than other sites (for example, our ALMA site has about 25% betterphase stability than our measurements from the VLBA site on Mauna Kea).

Yes, phase errors are really bad.The statistical nature of the phase fluctuations above Chajnantor is well understood – every

ten minutes, we determine the phase structure function from the site testing interferometerdata, which permits us to calculate the phase errors on any desired time or spatial scale, atany observing frequency. We have trouble extrapolating far beyond the 300 m baseline whichthe site testing interferometer measures, as the phase structure function is expected to flattensomewhere around 500 or 1000 m – depending upon the thickness of the turbulent atmosphericlayer. However, as we can see, if you are worried about what the phase structure function isdoing at 1000 m, you’ve already lost the war. All the action is down on short time and spatialscales.

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1.2 But Phase Errors Can be Corrected

Of course, we’ve been worrying intensely about the problem of phase correction on ALMAfor over a decade, and it is widely appreciated that we must win the battle for phases atthe shortest possible time scales. The two competing correction methods are water vaporradiometry (aka WVR; Hills and Gibson, 2001; Delgado 2001, Delgado et al, 2003), whichwill perform well on time scales as short as 1 s, but has problems tying in the fluctuationsto absolute water vapor levels (ie, has trouble with the long scales) – and fast switching,which, with typical switching time scales of 20 seconds, will effectively remove atmosphericstructure on time scales of about 10 seconds, and has no problems hooking into longer timescales. WVR’s problems on long time scales are due, not only to long time scale instrumentalstability, but also because WVR does not directly measure the water vapor, but must infer itfrom brightness temperature measurements via an inexact model; furthermore, the delay mustbe inferred, via a model, from the inferred water vapor.

The BIG hope is that these two seemingly complimentary phase compensation methodsmight work together as a coherent phase compensation strategy, but constructing and verifyingsuch a strategy requires either extensive simulations beyond our current capabilities, or anoperating ALMA with which to experiment. If we cannot use WVR to improve the shorttime scale fluctuations which fast switching cannot touch, we will need to deal with extensivecorrection for decorrelation, which will be both baseline and time dependent (Holdaway, 2003b,unpublished).

2 Overview of Fast Switching

2.1 Fast Switching with Calibration at the Target Frequency

Fast switching phase correction is a phase compensation strategy which solves for the antenna-based phases – which will be a combination of the atmospheric and instrumental phases – byquickly observing a moderately bright point source calibrator and quickly rushing back to thetarget source to observe for a limited integration. However, as the atmosphere changes and thecalibrator phase solutions go stale, we must go back and observe the calibrator again. With thenearly complete record of atmospheric phase stability data from Simon Radford’s site testingcampaign at Chajnantor, and our detailed understanding of the fast switching phase correctionprocess, we can calculate fairly precisely how often we must observe that calibrator and howmuch sensitivity we will lose by calibrating instead of observing, and how much sensitivitywill be lost due to decorrelation from the imperfectly calibrated phases. While there are manypopulations of sources which could serve as phase calibration sources, we think that the quasarswill provide a good combination of being both ubiquitous and unresolved.

Fast switching has been shown to work at several millimeter observatories. It is generallyaccessible to all telescopes, as it is little more than traditional phase and amplitude calibration,only done on time scales short enough to actually track some or most of the atmosphericfluctuations. However, as no telescope before ALMA has been designed with fast switchingin mind, on existing telescopes, we take what the telescope provides in terms of calibrationstrategies. On ALMA, we expect to see a highly tuned version of fast switching achieve themost we can, given what nature provides us in terms of the atmospheric conditions and thesky full of calibration sources.

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As a prelude to the next section, remember that fast switching at the target frequency mea-sures the sum of the instrumental and the atmospheric delays with every calibrator observationand requires nothing special.

2.2 Fast Switching Observing the Calibrator at a Lower Frequency

But ALMA will go yet another step beyond what existing telescopes are able to do: it is plannedthat ALMA will perform the phase calibration at the relatively low and sensitive frequencyof 90 GHz and scale the phases, presumed due to non-dispersive atmospheric delays (we treatdispersive delays later), up to the observing frequency. We’ll refer to this as dual-frequencyfast switching. Our past calculations of ALMA’s fast switching efficiency have appreciatedthe fact that the phase solutions at the lower frequency have to be very accurate in order toscale them to the target frequency. While we knew we would need to periodically observe abright calibrator at both the calibrator and target frequencies to solve for the instrumentalphases at both frequencies, we have never calculated the impact this so-called instrumentalsequence (also referred to as the cross-band calibration in some documents) would have on fastswitching efficiency. We needed an estimate of the brightness of the calibration sources as afunction of frequency, and it wasn’t until now that we have developed such estimates. In orderto more accurately predict the efficiency of fast switching phase correction on ALMA, we alsoneed to make progress in measuring the source counts of potential ALMA calibrators at higherfrequencies.

2.3 The Fast Switching Schematic

Following the lead of D’Addario (2003) in the analysis of the the instrumental sequence, wepresent a rather detailed picture of the sequence of observing operations that will be requiredof ALMA in dual-frequency fast switching phase calibration in Figure 1. A number of termsin this heuristic description of fast switching, some of which are important in the evaluationof fast switching in Section 7 are explicitly defined in Appendix A. This figure shows theinstrumental sequence (or cross-band calibration), a moderate number of target sequences(simple fast switching cycles, which is all we need if we observe the calibrator at the targetfrequency), and, before the instrumental phase has drifted significantly, a final instrumentalsequence. We call the instrumental sequence followed by several target sequences a single majorcycle.

The major cycle time is set by the instrumental stability time scale, which we assume in thiswork to be 300 s. While the simulations in this memo do not include any instrumental delayerrors, they will indicate how large the instrumental delay errors can be without seriouslydegrading ALMA’s performance while fast switching. Of course, we hope the instrumentalstability time scale is longer than 300 s, as it should be at moderate to low frequencies. If theinstrumental stability time scale is longer, our work is still valid as a lower limit for the overallfast switching efficiencies.

Conceptually, fast switching starts before the observations begin. We must have goodknowledge of the fluxes and positions of potential calibrator sources for both the target andinstrumental sequences. Given database of potential sources, possibly updated with recentobservations to provide current fluxes, we determine the optimal calibrators for both the targetsequence and the instrumental sequence (these calibrators will usually be different). It would

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take just over 1.5 s per potential calibrator to slew to and measure the flux of each potentialtarget sequence calibrator as the time is dominated by the slew time and these sources will tendto be of order a degree apart. The brighter, more distant instrumental sequence calibratorswill take longer to observe because they are farther away.

During a single 300 s major cycle, we perform a short (10-25s) instrumental sequence on abright (ie, 0.5 - 1.0 Jy) and relatively distant (5-15 deg) calibrator, followed by several (10-20)cycles of the target sequence, alternating between the target source and a very close (< 2 deg)moderately bright (0.05 - 0.2 Jy) phase calibrator. In the schematic, observations at the targetfrequency are unshaded, observations at the calibration frequency are shaded black, and slewingis shaded with diagonal lines. The majority of the time is spent on the target source at thetarget frequency. The calibrator observations in each target sequence cycle are very short, andthe instrumental sequence calibrator observations which bracket the 300 s major cycle are alsovery short.

In the instrumental sequence, we begin by slewing to the really bright calibrator sourcewhich we must accurately detect at both the cal and the target frequencies. A bright sourceminimizes the detection time and hence minimizes the residual atmospheric phase errors whichblow by in the time between calibrator and target frequency observations. With ALMA’ssensitivity, these sources can easily be detected with sufficient SNR at the calibration frequencyin 1 s. It will take 1.5 s to change to the target frequency. As we are racing the atmosphere andwe don’t have to change sky positions, any reduction in the time required to switch frequencies(ie, shorter than the 1.5 s specification) will be beneficial – in fact, the atmospheric error madein the instrumental sequence will ultimately set the requirements on the magnitude of theacceptable long-term electronic phase drift. Often, this calibrator source will also be brightenough to adequately detect the phase at the target frequency in a single 1 s integration, inwhich case we finish the instrumental sequence by slewing back and performing several targetsequences. If the instrumental sequence calibrator is not bright enough, we will need to switchback and forth between cal frequency observations and target frequency observations untilwe obtain high enough SNR on the phase solutions at the target frequency. We performthis frequency switching on the instrumental sequence calibrator to remove the contributionof atmospheric phase errors on time scales greater than about 2 s, essentially calibrating thetarget frequency observations of the instrumental calibrator with the cal frequency observationsof the same source.

In a single target sequence, (which was formerly all we considered in our model of a fastswitching cycle, and will sometimes be referred to in this work as a fast switching cycle), westart with a short calibration observation at the calibration frequency. The integration time onthe calibrator is determined by the calibrator’s strength, ALMA’s sensitivity, and the signal-to-noise ratio (SNR) required to scale the calibrator’s phase solutions to the target frequencyand still have errors which are less than the residual atmospheric phase errors. In all of oursimulations, this is always less than a second. The required SNR is set by the requirement thatthe thermal noise, converted into a phase at the target frequency, be small compared to theatmospheric phase contribution. This small thermal noise contribution to the phase is addedinto the final rms phase.

After we detect the phase on the calibrator with sufficient SNR, we slew from the calibratorto the target source. The specification for antenna motion is 1.5 deg in 1.5 s, which is a typicaldistance to a calibrator. Furthermore, the specification on the setup time to change frequenciesis 1.5 s, which will happen during the slew. In our modeling of the antenna slewing (Holdaway,

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300 s Major Cycle

10−20 Target Sequences

1 sec

f targetf cal f target f calslew slew

10−25 seconds for Instrumental Sequence

Target Source ObservationCalCal

15−30 seconds cycle time

slew slew

Instrumental Sequence (Cross Band Calibration) Single Target Sequence

Figure 1: Overview of fast switching, showing both the target sequence and the instrumentalsequence. The calibrator for the target sequence will typically be within 2 deg of the targetsource, while the calibrator used in the instrumental sequence will be up to 15 deg away fromthe target source.

2001 – with a correction by capping the maximum slew speed for long slews which will berequired for reaching the more distant instrumental sequence calibrator), it appears that veryshort slews (like 0.25 deg) might be completed in somewhat less than 1.5 s, so if we are able tochange frequencies in less than a second, we’ll take advantage of that capability here too. Afterslewing onto the target source with a specified pointing error of less than 3 acrsec (which willprobably not be acceptable for very high frequencies or mosaicing – more on this topic below),we begin the actual observations on the target source at the target frequency. After integratingon source at the target frequency for 10-25 s, which is determined by a global optimizationof the entire fast switching phase correction procedure considering the current atmosphericconditions, the characteristics of the optimal calibrator, and the cal and target frequencies,we again slew back to the calibrator, and as we reach the calibrator, here ends a single targetsequence.

Most often, we would immediately follow this target sequence with several more, startingon the phase calibrator again. If this is the last target sequence before another instrumental se-quence, we should throw in a final observation of the calibrator to facilitate phase interpolation.If we are about to end the source block, or go off and do another sort of calibration, we shouldthrow in a final instrumental sequence to facilitate in interpolating the scaled instrumentalphase difference.

2.4 What About Fast Switching While Mosaicing?

ALMA antennas have a 0.65 arcsec pointing error specification to permit accurate mosaicimaging and good single pointing sub-millimeter performance (sub-millimeter mosaicing willbe somewhat compromised by this pointing). So, what is the logic behind the 3 arcsec pointingspecification for fast switching? It was originally reasoned that fast switching would do little toimprove phase fluctuations on baselines shorter than 200 m (ie, vt/2 + d), and further reasonedthat most mosaicing would be performed in the compact configurations with baselines less than200 m. In these short configurations, the native phase stability of the atmosphere would permit

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observations without fast switching.However, since conceiving of this fast switching spec, ALMA has grown in collecting area,

and the smallest configuration is almost 200 m, while the effective switching scales can indeedbe less than 200 m. Furthermore, often mosaics will be made in larger configurations whichdo require phase calibration. And last, this specification was originally just a factor of 3 worsethan the mosaicing pointing spec of 1 arcsec for the MMA’s 8 m antennas – when we increasedALMA’s antennas to 12 m, we tightened the mosaicing pointing spec but forgot to tighten thefast switching pointing spec, so now fast switching’s pointing is a factor of 4.6 off of the staticpointing spec. So, what can we do?

We do not have a spec for how long it will take ALMA antennas to settle to the 0.65 arcsecpointing level after fast switching, so we will have to study the antenna’s settle down capabilitiescarefully to address this issue. Basically, when mosaicing and fast switching simultaneously,we will probably need to wait longer while the antennas slew to the target source to permitthem to achieve better pointing. How much better pointing and how long will depend uponthe required mosaic image quality, and in turn, on the mosaic’s SNR. We will often be able topermit larger pointing error on the calibrator, except at the higher frequencies. The time lostto waiting for the antennas to achieve superior pointing for mosaicing has not been included inthe analysis of the fast switching efficiency, but should be included after we have studied thisissue on the prototype antennas.

2.5 What About Fast Switching While Observing at Sub-millimeter Wave-lengths?

The same issue will cloud our observations of sub-millimeter wavelength sources. Even singlepointing sub-millimeter observations will require excellent pointing on the target source, somore time will be lost waiting for the antennas to settle to the desired pointing, which willdepend upon both the observing frequency and the required image quality.

3 Analysis of Fast Switching

3.1 RMS Phase Jitter, Phase Errors, and the Structure Function

To estimate the residual phase errors after fast switching, we have in the past used the expres-sion

σφ =√

Dφ(vatmostcycle/2 + d), (1)

where σφ is the rms residual phase error, Dφ is the phase structure function determined by thesite testing interferometer (Radford, 1996; Holdaway et al, 1995), vatmos is the velocity of theatmosphere at the height of the turbulent layer, tcycle is the fast switching cycle time, and dis the linear distance between the lines of sight to the target source and the calibrator, at thealtitude of the turbulent layer. This quantity represents the standard deviation of the residualphase within a single target sequence.

In the data analysis of the site testing interferometer data, we perform a power law fit tothe root structure function, and parameterize it as

√Dφ(ρ) = a · ρα,

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Figure 2: Most of the fast switching cycle will be closer to a calibration observation thantcycle/2. Hence, the phase errors will be somewhat less than

√Dφ(vt/2 + d); the d term is

generally small compared to vt/2.

where a is, in rough terms, the amplitude of the atmospheric phase fluctuations, and α is theroot phase structure function power law exponent. The median value of α, for six years of sitetesting at Chajnantor, is 0.58.

In the term vatmostcycle/2 + d, the d ends up being negligible: with vatmos=12 m/s andtcycle=20s, vt/2 will typically be 120 m. For a distance between calibrator and target sourceof 1 deg and a height of the turbulent layer of 500 m above the ground, d ends up being lessthan 10 m for a source at the zenith (lower elevation angles will make d somewhat larger).In the reasoning below, feel free to conceptually ignore d, although we did include d in thesimulations.

The expression in Equation 1 systematically overestimates the residual phase error whichwill occur in fast switching. Figure 2 illustrates the problem. The top graph shows the simulateduncorrected atmospheric phase time series for a 1000 m baseline interferometer, and the lowergraph shows the residual phase as a function of time on the target source after correcting withlinear interpolation. Some time is lost on the lower plot due to calibration observations andslewing. These plots only show the effects of the atmosphere, as the source visibility phase andinstrumental phases have not been included.

With yet another home grown AIPS++/glish tool, vtdsim.g, we have a laboratory toexplore the finer details of fast switching and better determine the relationship between the

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residual phase, the phase structure function, and the effective fast switching baseline vt/2 + d.From performing such detailed simulations of fast switching and the statistics of the resultingresidual phases, we have identified three effects which we need to address to more accuratelygauge the success of our full blown simulated fast switching observations which are presentedlater in this memo:

• The rms residual phase error, given by the rms of the residual phase time series (the lowergraph in Figure 2), will be a valid number for calculating decorrelation losses. However,the average phase error made on the target source visibility will be much smaller. Inthis case the rms residual phase is 25 deg, while the mean phase error on this integrationwill be -17 deg. In our analysis, we will consider both the rms residual phase error forpurposes of decorrelation calculations, and the rms of the average phases, for purposesof visibility error calculations.

• the rms phase error at the center of the target source integration (ie, 10 s, or tcycle/2)might appropriately be estimated by the root structure function evaluated at vt/2, butmost of the target source integration will be significantly closer to one or other calibra-tion observations. In Figure 2, the largest residual phase errors on the target sourceobservation occur between 8 and 13 s, and the phases near the calibrations are muchsmaller.

• For a root structure function exponent α of 0.58, linear interpolation will provide amarginal improvement over a “closest calibrator” interpolation. For flatter α (ie, morelike white noise), linear interpolation could actually be worse than closest calibratorinterpolation, and for steeper α (ie, up to the maximum theoretical value of 0.83), linearinterpolation should significantly reduce the residual phase. Hence, the interpolationmethod should depend upon the details of the atmospheric conditions.

Including these three factors in simulations (mean vs rms phase, interpolation improvements,and vt/2 being the worst case), we simulated fast switching observations through a modelatmosphere with linear interpolation, and we empirically determine that the rms phase erroris indeed less than the the phase structure function evaluated at vt/2 + d, reduced by a factorγ0(α) (we have not demonstrated any dependence upon α, but it is reasonable):

σφ = γ0(α) ·√

Dφ(vatmostcycle/2 + d), (2)

and the rms of the phase errors averaged over each target integration can be written as thephase structure function reduced by γ1(α):

σφ̄ = γ1(α) ·√

Dφ(vatmostcycle/2 + d). (3)

We expect the γ factors to be weak functions of α, though at this time we have not performedexhaustive studies. For the simulations we performed, with α = 0.58, and tcycle = 20s, we findthat γ0 = 0.65 and γ1 = 0.35

If you observe with a single interferometer through the atmosphere with phase errors, therms phase (after subtracting out the average phase) will increase with integration time until,after many “crossing times” (ie, the baseline length divided by the atmospheric velocity), therms phase will saturate. When we worked out the site testing interferometer data reduction

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pathway (Holdaway et al, 1995), we performed this analysis, looking at how the rms phaseincreased with integration time. The rms phase reached a plateau at about 10 minutes, andthen started to increase again for longer integration times. We interpreted this to mean that 10minutes integration resulted in a saturated phase, and beyond 10 minutes, phase errors fromthermal changes in the cables or satellite motion came into play.

However, our simulations with vtdsim.g indicate that a 600 s integration on a 300 mbaseline and a 12 m/s wind is not really enough to reach saturation. When simulating a 300 mbaseline interferometer with the vtdsim.g machinery, we find that the rms phase only accountsfor about 0.70 of the phase structure function at 300 m. It is unclear if reanalysis of the sitetesting interferometer data using a longer integration time is the right thing to do, as non-atmospheric phase errors will creep in. For now, there is a very simple fix we can apply, andthat is to incorporate this correction factor, 1/0.70, into the γ factors:

γ′0 = γ0/0.70 = 0.93

γ′1 = γ1/0.70 = 0.50

One more detail we must address is the way the phase errors will average down over severalfast switching cycles. That γ ′

1 is 0.50 indicates that the standard deviation of the averagephases over several switching cycles will be significantly lower than the rms phase. Typically,we will perform 10-20 such fast switching cycles, and, as the residual atmospheric errors afterfast switching calibration are random, and the noisy gain errors on the fast switching calibratorare random, averaging the target source visibilities from several fast switching cycles will resultin better and better estimates of the visibility phase. We assert that the ensemble average ofthe mean phase error after Ncycles fast switching cycles will go approximately as

〈φ̄〉 = γ′1(α)/

√Ncycles ·

√Dφ(vatmostcycle/2 + d).

This assertion relies upon the Ncycles being independent. The crossing time of long baselineswill be much longer than the target sequence cycle time, so each switching cycle will not bestrictly independent. However, the fact that new atmosphere is coming into view (at leaston one antenna) with each new cycle, and the randomizing influences of interpolation, shouldresult in the the average phases for each cycle being largely independent.

One final point concerning the number of visibilities which can be averaged is the crossingtime of a visibility across a Nyquist sampled (u,v) cell due to earth rotation. The size of the(u,v) cells is set by the inverse of the field size we are imaging, and the number of cells will beproportional to the baseline length. Long baselines will zip across the (u,v) cells very quickly,short baselines will move through the cells slowly. The crossing time sets an upper limit to theamount of time one can average the visibilities for.

For full primary beam imaging, the (u,v) cell crossing times are given in Table 1.

3.2 Error Analysis of Fast Switching

We start here by analyzing the phase errors made in fast switching in the case that the cali-bration is performed at the same frequency as the target source is observed at. In this case,no instrumental sequence calibration needs to be performed. The phase jitter, appropriate forcalculations of decorrelation, is then given by Equation 2, and the mean phase error in a singlefast switching cycle is given by Equation 3.

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Baseline [m] tint [s] Ncycles γ′1/

√Ncycles

200 412 20 0.11400 206 10 0.161000 82 4 0.252000 41 2 0.354000 20 1 0.510000 8 - -

Table 1: The (u,v) cell crossing time for full primary beam imaging, for a baseline of thespecified length, and the approximate number of fast switching cycles which could be averagedand the reduction factor in the phase error due to averaging. If only a portion of the primarybeam were being imaged, the (u,v) cell sizes would be larger, the crossing time longer, andmore averaging could take place, so these numbers represent a worst case.

Next, following D’Addario’s memo (D’Addario, 2003), we analyze the errors made in fastswitching for the case where the target frequency νt and the calibration frequency νc aredifferent.

3.2.1 Target Sequence

By scaling the calibrator phase φcal by the ratio of the frequencies and subtracting that fromthe target phase φtar (ie, we are ignoring dispersive effects), we get

φtar −(

νt

νc

)φcal = φtrue

tar + φi(νt) −(

νt

νc

)φi(νc) + γ12πνt

√Dτ (vtt.s./2 + d) +

(νt

νc

)εc (4)

where φtruetar is the true target source visibility at the target frequency, which can be ignored in

this analysis, the φi terms are instrumental phases (assumed here to be constant, or at leastsmall compared to other problematic phases), Dτ (vtt.s./2 + d) is the delay structure function(ie, the phase structure function, cast into a frequency-independent form as delay), tt.s. is thecycle time for the target sequence, and εc is the phase error on the calibrator gain solutiondue to thermal noise. φtar is the phase of a single baseline’s visibility, but to reduce thenoise contribution, φcal is synthesized by solving for the antenna-based gains at the calibrationfrequency (ie, an improvement in SNR by

√N − 2), and then differencing two antenna-based

phases (ie, a loss in SNR of√

2 to result in an overall gain of√

(N − 2)/2) to give the highSNR estimate of the calibrator visibility phase at the calibration frequency.

3.2.2 Instrumental Sequence

We define the scaled instrumental phase difference Φi to be

Φi = φi(νt) −(

νt

νc

)φi(νc). (5)

We can estimate Φi by performing a quick two-frequency observation a calibrator source whichis bright at both the target and calibrator frequencies and forming the scaled phase difference,but this estimate will be contaminated by both atmospheric phase errors which occur on that

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short time scale while the observing frequency (but not the source) changes, and thermal noisedue to the short observations intended to minimize the atmospheric errors:

Φ̂i = φtar −(

νt

νc

)φcal (6)

= Φi + γ12πνt

√Dτ (vti.s./2) +

√√√√ε2t +(

νtνc

)2ε2c

Ni.s., (7)

where ti.s. is the time between the center of the target frequency observation and the center ofthe cal frequency observation in the instrumental sequence (taken in our analysis to be 2 s),and Ni.s. is the number of instrumental sequence integrations that must be performed to reachthe required SNR. εt is the phase error due to noise in the gains at the target frequency.

How much time do we need to detect Φ̂i, or how do we calculate the noise in the finalterm? We need to synthesis each baseline-based Φ̂i from the antenna-based gains at the twofrequencies, as we did in the target sequence, to get higher SNR. However, we may still needmore SNR on Φ̂i. To get that SNR, we can’t just integrate longer, because then we would bedominated by the atmospheric residuals. To minimize the atmospheric phase errors, we usevery short (ie, 1 s) integrations at the calibration and target frequencies and change betweenthe frequencies as fast as we can (the specification is 1.5 s, and we are hoping for 1 s). Thecycle time for the instrumental sequence measurement, ti.s., will be twice the integration timeplus twice the frequency change time. We can remove the effects of the atmosphere on timescales of ti.s./2 % 2 s by forming the scaled phase difference, or Φ̂i, with the lower-than-desiredSNR phase data that we get from a single pair of 1 s measurements at the two frequencies.If we need higher SNR, then we need to perform this measurement cycle multiple times andaverage each value of Φ̂i until the SNR improves sufficiently. In order to keep the noise inthe scaled phase difference well behaved, we should probably represent Φ̂i as the phase of acomplex number, and average the complex numbers, and the take the final phase.

We assume that the atmospheric phase errors and the thermal noise are statistically uncor-related, so the rms error phase fluctuations are given by adding the atmospheric term and thenoise term in quadrature.

4 Calibrator Source Count Estimates

In order to calculate the tcycle to evaluate fast switching, we need to know tcal (the integrationtime on the calibrator) and tslew (the slew time to and from the calibrator), so we need toknow how bright and how far away the calibrator sources are likely to be.

4.1 What Sources Are Possible Calibrators?

The success of fast switching is dependent upon an adequate supply of moderately brightcalibration sources close by to the target source. The main population of calibrators for fastswitching is the flat spectrum quasars. SiO masers have been suggested, but there are notenough of them which are bright enough to make a significant impact on our need for calibrators(Holdaway, 1996). Ultra compact HII regions have also been considered, but they are extendedand there probably aren’t enough either (Holdaway, 1996, unpublished). Recently, Butler(2003) has suggested using thermal dust emission from distant galaxies at high frequencies

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where they will dominate the sky (rather than calibrate sub-millimeter observations at 90GHz), but these sources will be extended (approximately 1 arcescond), and are probably notoptimal for phase calibrators. Hence, we restrict the analysis of calibrators to the flat spectrumquasars. Adding any other population of sources to the calibrator pool will only improve theefficiency of fast switching, so our work will stand as a valid lower limit to fast switching’sperformance.

Butler’s work does raise an interesting question. If we perform the fast switching phasecorrection at the same frequency as the target source observations, the target sequence solvesfor both the atmospheric and instrumental delays, and we bypass the instrumental sequence(and any associated errors). Logically, at low target frequencies where the quasar fluxes arehigh and the ALMA sensitivity is good, it should be optimal to perform the fast switchingphase calibration at the target frequency. At high target frequencies where the quasar fluxesare falling and ALMA sensitivity is poor, it will be advantageous to perform the fast switchingcalibration at a lower frequency and scale the solutions to the target frequency. So, there willbe a frequency below which we will perform the phase calibration at the target frequency, andabove which we will perform the phase calibration at 90 GHz and scale the solutions up. Whatis that magic frequency?

To answer this, as well as to investigate the instrumental sequence, we need to estimate thequasar source counts at each ALMA band.

4.2 Estimating Flat Spectrum Quasar Source Counts in the Millimeter andSub-millimeter

Figure 3 shows the histogram of the spectral index (S(ν) ∝ ν−α) distribution from 8 GHz to90 GHz for the 367 flat spectrum quasars observed by Holdaway, Owen, and Rupen (1994).(Our apologies for reusing the symbol α for both the quasar spectral index and the atmosphericstructure function exponent, but the context should indicate which concept we are using.) The8 GHz core fluxes were measured by Patnaik et al. with the VLA in A array. The 90 GHzfluxes were measured at the NRAO 12 m. Under the assumption that any extended emission issteep spectrum and will be negligible at 90 GHz, we posit that this spectral index distributiondescribes the quasar cores. We found that the spectral index distribution did not depend uponthe quasar flux, so we used all 367 quasars to determine a single spectral index distributionusing Feigelson’s ASURV code (Feigelson, 1985).

We have also performed a two Gaussian fit to this histogram (dotted lines, the sum of whichis the dashed line). The black boxes represent the integral of the summed Gaussian fit over thehistogram boundaries, appropriately normalized for comparison to the histogram. Because thenumber of degrees of freedom in the fit (6) is about the same as the number of independent datapoints in the histogram (7), we expect a very good fit. We have used the Gaussian fit, ratherthan the 7 bin histogram, for our extrapolations of source counts to the higher frequencies, asthe sharp edges were causing some odd bumps in the source counts extrapolations.

Starting with the source counts of flat spectrum quasars at 5 GHz (Condon, 1984), wemodulated the fluxes of these counts by the core fraction distribution of the 3CR2 flat spectrumsources, so that the modified counts represented the source counts of only flat spectrum cores,ignoring extended components. We further modulated the source counts by the flat spectrumquasar spectral index distribution between 8 GHz and 90 GHz which we determined, followingthe prescription by Condon (1984) to scale the source counts up to 90 GHz.

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Figure 3: The seven bin histogram is the spectral index (S(ν) ∝ ν−α) distribution from 8 GHzto 90 GHz for the 367 flat spectrum quasars observed by Holdaway, Owen, and Rupen (1994).See the text if you want to know about the Gaussians and filled squares.

At frequencies above 90 GHz, the quasar cores are expected to become optically thin,which will result in a spectral steepening. Hence, for frequencies above 90 GHz, we assumethat the sources have a spectral break and we steepen the entire spectral index distributionby 0.5 when estimating quasar source counts at frequencies above 90 GHz. This steepeningresults in a median spectral index of 0.8 at frequencies about 90 GHz, which appears to bereasonable to several famous radio astronomers (Frazer Owen, Richard Hills, and Robert Laing;private communications). The resulting integral source count estimates, including the spectralsteepening above 90 GHz, are displayed for a range of frequencies in Figure 4.

We have written an AIPS++/glish tool, sourcecountsim.g, which performs the frequencyscaling of the source counts and can simulate a calibrator field which is consistent with theestimated source counts and spectral index distribution. The sourcecountsim.g code is notpart of the AIPS++ distribution, but can be obtained from Mark Holdaway’s home pageat http://www.tuc.nrao.edu/ mholdawa/, along with other software used to perform thecalculations in this memo.

5 What Frequency Should We Calibrate At?

As mentioned above, the default scheme for fast switching is that we will calibrate at 90 GHzand scale the phase solutions up to the observing frequency. There are two issues which pertainto the assumption of calibrating at 90 GHz which we need to address now.

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Figure 4: These are the integral source counts of flat spectrum quasars at various ALMAfrequencies. The counts at 90 GHz will be fairly accurate, but we are guessing at the spectralindex distribution for higher frequencies: to estimate the higher frequency source counts, we aretaking the measured 8-90 GHz spectral index distribution and steepening it by 0.5, resultingin a median spectral index of 0.8 above 90 GHz.

5.1 Is 90 GHz the Optimal Calibration Frequency?

We have not examined this in detail yet (!), and we should, just to cover our bases. If dual-frequency fast switching is to be used, band 3 is probably the best frequency for the calibration.We have investigated which frequency is optimal for pointing calibration at the ATF (Holdawayand Mangum, 2001), and the combination of the decent sensitivity and the “lever arm” ofthe smaller beam at the higher frequency makes 90 GHz superior to 30 GHz for pointingdeterminations. There are similar arguments for fast switching phase correction that have notbeen entirely spelled out.

What about performing the calibration observations at higher frequencies, like 140 GHz,and scaling up to the target frequency? Answering this question would probably be dominatedby our assumptions about how the calibrator fluxes are falling down, and it is probably notworth investigating this question at this time.

When the final ALMA systems are in production and we understand the system temperatureas a function of frequency, we can determine the optimal frequency band within band 3 for fastswitching.

5.2 Potential Issues with Instrumental Sequence Calibration Sources

We have asserted that the instrumental sequence will be performed on a quasar which isbright at both the calibrator frequency and the target frequency. The success of this strategydepends, among other things, on the quasar looking the same at the calibration and the target

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frequencies. This may not be the case.For example, as dust emission increases in flux with frequency, the quasar point source

may be surrounded by extended dust emission. The dust emission problem probably doesn’texist for moderate frequencies, perhaps only for frequencies of 400 GHz and above. If thedust emission is symmetric and centered on the quasar, it won’t contribute to the phases.Furthermore, if we are observing in a very large configuration, much of the dust emission couldbe resolved out – in other words, a (u,v) range could be determined observationally in whichthe calibration source was well approximated by a point source. Such a (u,v) range wouldeliminate some baselines from the phase solution, thereby requiring a brighter calibrator ormore integrations on the calibrator.

Another potential problem is that the position of the quasar core will depend upon thefrequency. As we observe at higher frequencies, the region at which the jet becomes opticallythick (functionally “the core”) will be closer and closer to the central engine. If the posi-tion difference with frequency is large enough, this would introduce a systematic error in theinstrumental sequence. We need to investigate how large this systematic error might be.

5.3 Dispersive Phase, Dry Fluctuations, and Implications on the CalibrationFrequency

We claim that we can scale the phase solutions from the calibrator frequency up to the targetfrequency simply by multiplying by the ratio of the frequencies. This is only true for targetfrequencies at which the atmosphere is non-dispersive, or up to about 350 GHz. However,at sub-millimeter wavelengths, the atmosphere is dispersive enough to result in significantdeviations from this simple phase scaling (Holdaway and Pardo, 2001). At the very least,we will be required to employ radiation transmission codes such as ATM to instruct us inthe proper phase scaling from calibrator to target frequencies. However, we could be limitedin the accuracy of the transmission codes, or we may be limited in our knowledge of physicalatmospheric parameters such as the temperature profile or the water vapor profile, which wouldresult in some errors in our prescribed phase scaling. We can study the latter question nowusing ATM with various atmospheric profiles. In this memo we do not attribute any errors tothe phase scaling process.

However, we must consider the possibility that the dispersive phase in the sub-millimeterrange may not permit fast switching to accurately scale the phases from calibration to targetfrequency. If this were the case for certain frequencies or certain atmospheric conditions, wewould have to resort to performing the fast switching calibration at the target frequency andmake do with the weaker or more distant phase calibrators. But again, the calibrators mayhave dust structure at the higher frequencies and may not be appropriate for calibrating at thetarget frequency when observing in the sub-millimeter. Later in this memo, we document howeffective fast switching is at each target frequency, and it is not much worse than calibratingat 90 GHz.

While the dry component of the atmosphere results in the largest portion of the overallatmospheric delay, we expect that the dry atmosphere will be relatively stable and will con-tribute essentially none of the phase fluctuations. If there are significant phase fluctuationsintroduced by fluctuations in the dry atmosphere, fast switching with different calibration andtarget frequencies will still work fine unless there is a significant dispersive phase. In otherwords, if there are dry phase fluctuations, we can calibrate at 90 GHz and scale the phase up

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to about 350 GHz, but we we can’t scale the phase up to the sub-millimeter where the phase isdispersive – basically, we would need to know how much of the phase errors are dry and non-dispersive and how much of the phase errors were wet and dispersive. So, if dry fluctuationswere significant, sub-millimeter observations using fast switching phase correction would needto observe the calibrator at the target frequency.

6 Our Method of Calculating the Residual Phase Errors

6.1 This is a “squishy” sort of problem

Fast switching is a very complex multi-dimensional system. The variables that are out thereinclude: the inherent sensitivity of the telescope, the distance to the calibrator, the strength ofthe calibrator, how much time we spend calibrating (multiply the last three dimensions by two,as there are both the target sequence calibrators and the instrumental sequence calibrators),how much time we spend observing our target source before we go back and observe thecalibrator again, the nature and level of the atmospheric phase fluctuations, and the velocity ofthe turbulent layer. With so many dimensions, the overall fast switching system becomes rather“ floppy” – if one dimension of the problem becomes much harder (say we discover that the 90GHz calibration sources are only half as bright as we had assumed), the impact of that changebecomes absorbed in several dimensions (we go a little bit further away to reach a somewhatbetter calibrator, integrate a bit longer, shorten the overall cycle time, which decreases theon-target-source duty cycle), resulting in a final result, the overall observing efficiency, whichis only marginally reduced.

The one exception to this seems to be the atmosphere. If you make the atmosphere twiceas bad, the lost observing efficiency (due to both time lost to calibration and to phase decor-relation) gets twice as large.

6.2 Monte Carlo Simulations

Our job is to estimate how well fast switching phase correction will work. This is a messy,multi-dimensional problem, and we must consider

• varying and largely uncorrelated atmospheric opacity and phase fluctuations

• observations at frequencies ranging from 35 to 900 GHz

• variation of the quality of the calibrator field around the target source: some targetswill have fortuitously bright quasars very close by, other targets will not have a goodcalibrator very near. Additionally, each calibrator source will have a different spectralindex, so some really good calibrators at 90 GHz will be useless for the instrumentalsequence.

• variation in the velocity of the moist turbulent layer, which we ignore by taking it to be12 m/s.

• variation in the height of the turbulent layer, which we simply take to be 500 m. Theturbulent height only comes into play with the term d, the linear distance between thelines of site to the cal and target sources at the height of the turbulence. As d ends upbeing essentially negligible, any error in this height will have very little effect.

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Freq Band tau phase[GHz] at 225 GHz [deg]

unused 0 0.101 11.4443 1 0.220 6.7180 2 0.129 5.4490 3 0.129 3.635

145 4 0.089 2.955190 5 0.060 3.210230 6 0.056 2.175345 7 0.045 1.840500 8 0.037 1.615680 9 0.032 1.155880 10 0.025 0.790

Table 2: Median values of the 225 GHz opacity and the 11.2 GHz, 300 m baseline site testinginterferometer rms phase allocated to the ten ALMA observing bands.

As we have done in the past (Holdaway, 2001; Holdaway 2003), we reduce the dimensionalityof this problem by matching the observing frequency to the atmospheric conditions to simulatethe results of dynamic scheduling. Unlike Holdaway 2001, but similar to Holdaway 2003, weconsider both the opacity and the phase stability in splitting up the atmospheric conditionsamong the observing bands. So, while the best opacity and stability conditions from the jointdistribution will go to band 10, we have fallen short of a fully-optimized analysis (ie, pickingthose remaining atmospheric conditions that truly optimize the sensitivity at each observingband). We have thrown out the worst 20% of observing conditions as unusable (in retrospect,this is a bit extreme, and we could have thrown out 10-15%), and have assumed that each bandwill be used for 8% of the time. The median values for the 225 GHz opacity and 11.2 GHz/300msite testing interferometer for the conditions allocated to each observing band are shown inTable 2.

The only remaining dimensions of the problem we must worry about are the calibratorfluxes, the proximity of the calibrators to the target source, and the calibrators’ spectral indexes.(In general, we have to find two calibrators for each target source, one for the target sequence,and one for the instrumental sequence.) We deal with the calibrator properties by performingMonte Carlo simulations of representative calibrator fields 40 deg by 40 deg. The source countsin each calibrator field is consistent with the 90 GHz source count estimates we address above,and the spectral index distribution is the one measured from 8 to 90 GHz, steepened by 0.5above 90 GHz. We have created 1000 such calibrator fields, and we use the same fields for eachdifferent observing frequency or atmospheric conditions so that the fast switching results don’tdepend upon variations in the underlying calibrator source fields. For each observing scenario(ie, target frequency, calibration frequency, and atmospheric conditions), we simulate a fastswitching observation for each of the 1000 calibrator fields, selecting the calibrator sourceswhich result in the maximum efficiency for the fast switching observation. This process givesrise to 1000-element distributions of optimal target sequence calibrator flux, optimal efficiencies,optimal residual phase errors, etc. Median values of these distributions are reported below.

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Using the same simulated calibrator fields for each different observing scenario will havethe affect of reducing noisy scatter in the results among the different target bands so relativeconclusions are not affected, but will also increase any systematic errors as the entire ensembleof simulations will suffer from them – but with 1000 different instantiations of the millimetercalibrator sky, the systematics should be small.

The simulations which we report on here were carried out in AIPS++/glish which a suiteof tools developed for ALMA work:

• drivefast.g drives the simulation tool for the specific cases we studied and implementsthe Monte Carlo aspect of the simulations.

• fastswitchsim2.g encodes the essential logic of selecting the optimal calibrator andcalculating the residual phase errors and efficiencies.

• sourcecountsim.g simulates fields of calibrators consistent with specified source countstatistics and spectral index distribution, and also shifts source counts in frequency.

• stidata.g encapsulates transforming the site testing data to other frequencies and base-lines.

• almasensitivity.g is used to calculate the noise level of a certain observation or theamount of time that is required to reach a certain noise level.

• almatau.g encapsulates the opacity characteristics of the atmosphere above the ALMAsite.

These glish tools can be found at http://www.tuc.nrao.edu/ mholdawa/

7 Results

We present the quantities that are defined in Appendix A, as calculated with our Monte Carlosimulations, in Tables 3 through 5. The γ ′ factors have been included in calculating theseresults. Table 6 deals with the 5th percentile atmospheric phase and opacity, and is specificallyincluded so we can derive specifications for the electronic phase fluctuations. These tables showthe median values of the various parameters calculated in the simulations.

The units of the quantities in these Tables are: targetfreq, calfreq [GHz], el [deg], rms300(site testing phase) [deg], phases and gainerror [deg], calflux [Jy], caldist [deg], tslew, tcal,tcycle, ttarget, toverhead [seconds]. Efficiencies are noise-like quantities, as opposed to time-like quantities. The overal efficiency, eff0, is the product of the eff1, the square root of the dutycycle, and eff2, which id just the decorrelation from residual phase errors.

These tables are sort of sideways so many relevant terms can be shown. There are a numberof interesting results:

• Things are really very demanding in the sub-millimeter, with typical overall efficiencies(all eff0) of about 0.70; total residual phase fluctuations are between 30 and 40 deg. Meanphase errors are about half of this.

• At millimeter wavelengths, the efficiencies are more like 0.78 and up to about 0.90 at thelowest frequencies; total residual phase fluctuations are between 20 and 30 deg. Meanphase errors are about half of this.

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Figure 5: Histogram of the residual phase rms for the simulations reported on in Table 6 at875 GHz.

• Quite surprisingly, dual-frequency fast switching is only marginally more effective thancalibrating at the target frequency. Up to a target frequency of 345 GHz, calibrating atthe target frequency is as efficient or more efficient than calibrating at 90 GHz, scalingthe phases, and taking the extra time to perform the instrumental sequence.

So the reader can get a better feel for the sort of variation in the fast switching resultsdue to the Monte Carlo simulations (which deals only with the randomness of the calibratorfield), we show histograms of the 1000 results for the case of 875 GHz. Figure 5 shows thehistogram of the total residual phase (87% of the simulations resulted in rms phases between32.5 and 40 deg, and the median was 36 deg), and Figure 6 shows the histogram of the morenarrowly peaked total efficiency for the fast switching process (87% of the simulations resultin efficiencies between 0.65 and 0.75, with a median value of 0.69).

One surprising result that we can glean from this set of simulations is that fast switchingwill work pretty well even when calibrating at the target frequency. In fact, Calibrating at thetarget frequency should be almost as good as performing the calibration at 90 GHz and scalingto the target frequency, for target frequencies as high as 345 GHz. A marginal efficiencyimprovement is obtained at higher target frequencies – which, unfortunately, might requirecalibrating at the target frequency if there are problems with calculating the dispersive phase.

In retrospect, fast switching with the 90 GHz calibration frequency was designed for theMMA with 2000 m2 collecting area. Now, with a much larger ALMA telescope and over

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targetfreq 850 850 650 650 490 490 [GHz]calfreq 90 850 90 650 90 490 [GHz]el 60 60 60 60 60 60 [deg]tau225 0.032 0.032 0.037 0.037 0.045 0.045rms300 0.79 0.79 1.155 1.155 1.615 1.615 [deg @ 11.2GHz]all phase 31.53 36.70 33.09 37.86 33.77 38.78 [deg]phase drift 9.073 6.940 8.974 6.630 9.208 6.626 [deg]all eff0 0.749 0.691 0.734 0.677 0.727 0.666all eff1 0.872 0.845 0.868 0.845 0.865 0.845all eff2 0.859 0.814 0.846 0.803 0.840 0.795i.s. eff0 0.963 1 0.966 1 0.964 1i.s. eff1 0.972 1 0.975 1 0.975 1i.s. eff2 0.991 1 0.991 1 0.990 1t.s. eff0 0.777 0.691 0.760 0.677 0.754 0.666t.s. eff1 0.894 0.845 0.894 0.845 0.894 0.845t.s. eff2 0.867 0.814 0.854 0.803 0.849 0.795t.s. calflux 0.141 0.063 0.109 0.072 0.090 0.085 [Jy]t.s. caldist 1.927 3.589 1.659 3.472 1.527 3.440 [deg]t.s. tslew 1.724 2.360 1.592 2.295 1.529 2.273 [s]t.s. tcal 0.447 4.746 0.381 3.622 0.318 3.436 [s]t.s. tcycle 21.23 31.69 18.56 27.17 17.07 25.87 [s]t.s. Ncycles 13.36 9.463 15.38 11.03 16.70 11.59t.s. ttarget 17.06 22.86 14.60 18.95 13.52 17.76 [s]t.s. toverhead 4.184 8.941 3.851 7.774 3.595 7.599 [s]t.s. phase 30.57 36.70 32.16 37.86 32.73 38.78 [deg]t.s. atmophase 30.16 36.35 31.76 37.53 32.35 38.46 [deg]t.s. gainerror 6.794 8.155 7.146 8.415 7.274 8.619 [deg]i.s. calflux 1.136 -1 1.027 -1 1.124 -1 [Jy]i.s. tarflux 0.326 -1 0.325 -1 0.402 -1 [Jy]i.s. caldist 11.08 0 9.608 0 9.978 0 [deg]i.s. tslew 5.546 0 4.938 0 5.085 0 [s]i.s. tcal 4 0 4 0 4 0 [s]i.s. tcycle 300 300 300 300 300 300 [s]i.s. ttarget 283.7 300 285.5 300 285.1 300 [s]i.s. toverhead 16.22 0 14.41 0 14.80 0 [s]i.s. phase 7.658 0 7.602 0 7.941 0 [deg]i.s. atmophase 5.264 0 5.886 0 6.204 0 [deg]i.s. gainerror 5.569 0 4.810 0 4.957 0 [deg]same cal 0.052 0 0.056 0 0.042 0

Table 3: Median fast switching results for 850, 650, and 490 GHz, giving the best conditions tothe high frequencies. (Since the median values of each distribution are reported, times or phaseerrors might not add up exactly as expected. Also, phase errors are for one baseline.) Notethat the second (fourth, sixth) columns are for the same target frequencies, but performingfast switching calibration at the target frequencies instead of 90 GHz. “t.s.” quantities arefor an individual target sequence, “i.s.” quantities are for the instrumental sequence, and “all”quantities refer to products of “t.s” and “i.s.” quantities.

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targetfreq 345 345 230 230 190 190 [GHz]calfreq 90 345 90 230 90 190 [GHz]el 60 60 60 60 60 60 [deg]tau225 0.045 0.045 0.056 0.056 0.06 0.06rms300 1.84 1.84 2.175 2.175 3.21 3.21 [deg @ 11.2GHz]all phase 28.90 30.03 24.68 25.05 27.52 28.06 [deg]phase drift 7.055 4.852 5.919 4.281 6.582 4.385 [deg]all eff0 0.788 0.782 0.834 0.840 0.807 0.807all eff1 0.897 0.894 0.916 0.925 0.906 0.912all eff2 0.880 0.871 0.911 0.908 0.891 0.886i.s. eff0 0.979 1 0.983 1 0.982 1i.s. eff1 0.985 1 0.986 1 0.986 1i.s. eff2 0.995 1 0.997 1 0.995 1t.s. eff0 0.805 0.782 0.849 0.840 0.821 0.807t.s. eff1 0.912 0.894 0.930 0.925 0.919 0.912t.s. eff2 0.884 0.871 0.913 0.908 0.894 0.886t.s. calflux 0.079 0.061 0.071 0.050 0.057 0.053 [Jy]t.s. caldist 1.414 2.193 1.327 1.666 1.155 1.586 [deg]t.s. tslew 1.487 1.825 1.449 1.588 1.371 1.552 [s]t.s. tcal 0.283 0.538 0.216 0.327 0.177 0.325 [s]t.s. tcycle 20.15 23.13 24.13 25.88 19.79 21.65 [s]t.s. Ncycles 14.45 12.96 12.09 11.59 14.75 13.85t.s. ttarget 16.70 18.65 20.82 22.06 16.67 17.88 [s]t.s. toverhead 3.441 4.499 3.282 3.779 3.112 3.693 [s]t.s. phase 28.32 30.03 24.33 25.05 27.03 28.06 [deg]t.s. atmophase 27.87 29.61 23.81 24.54 26.56 27.61 [deg]t.s. gainerror 6.294 6.674 5.407 5.567 6.007 6.236 [deg]i.s. calflux 0.338 -1 0.219 -1 0.197 -1 [Jy]i.s. tarflux 0.130 -1 0.106 -1 0.110 -1 [Jy]i.s. caldist 3.782 0 2.864 0 2.603 0 [deg]i.s. tslew 2.395 0 2.093 0 2.008 0 [s]i.s. tcal 4 0 4 0 4 0 [s]i.s. tcycle 300 300 300 300 300 300 [s]i.s. ttarget 291.2 300 291.8 300 291.9 300 [s]i.s. toverhead 8.790 0 8.186 0 8.017 0 [s]i.s. phase 5.566 0 4.298 0 5.154 0 [deg]i.s. atmophase 4.977 0 3.922 0 4.781 0 [deg]i.s. gainerror 2.492 0 1.757 0 1.922 0 [deg]same cal 0.217 0 0.307 0 0.291 0

Table 4: Median fast switching results for 345, 230, and 190 GHz, giving the best conditionsto the high frequencies. Note that the second (fourth, sixth) columns are for the same tar-get frequencies, but performing fast switching calibration at the target frequencies instead of90 GHz. “t.s.” quantities are for an individual target sequence, “i.s.” quantities are for theinstrumental sequence, and “all” quantities refer to products of “t.s” and “i.s.” quantities.

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targetfreq 145 145 90 70 35 [GHz]calfreq 90 145 90 70 35 [GHz]el 60 60 60 60 60 [deg]tau225 0.089 0.089 0.129 0.129 0.22rms300 2.955 2.955 3.635 5.44 6.71 [deg @ 11.2GHz]all phase 22.28 22.27 18.66 20.46 15.38 [deg]phase drift 5.312 3.937 3.711 3.784 3.610 [deg]all eff0 0.860 0.870 0.906 0.889 0.934all eff1 0.928 0.939 0.955 0.948 0.969all eff2 0.927 0.927 0.948 0.938 0.964i.s. eff0 0.984 1 1 1 1i.s. eff1 0.987 1 1 1 1i.s. eff2 0.998 1 1 1 1t.s. eff0 0.874 0.870 0.906 0.889 0.934t.s. eff1 0.939 0.939 0.955 0.948 0.969t.s. eff2 0.928 0.927 0.948 0.938 0.964t.s. calflux 0.056 0.048 0.042 0.059 0.033 [Jy]t.s. caldist 1.153 1.329 1.029 1.117 0.753 [deg]t.s. tslew 1.363 1.450 1.288 1.345 1.137 [s]t.s. tcal 0.161 0.197 0.119 0.143 0.095 [s]t.s. tcycle 26.81 27.60 32.86 29.41 41.84 [s]t.s. Ncycles 10.89 10.86 9.127 10.19 7.169t.s. ttarget 23.74 24.32 29.97 26.40 39.30 [s]t.s. toverhead 3.079 3.273 2.858 3.021 2.537 [s]t.s. phase 21.98 22.27 18.66 20.46 15.38 [deg]t.s. atmophase 21.41 21.71 17.98 19.84 14.55 [deg]t.s. gainerror 5 5 5 5 5 [deg]i.s. calflux 0.148 -1 -1 -1 -1 [Jy]i.s. tarflux 0.099 -1 -1 -1 -1 [Jy]i.s. caldist 2.246 0 0 0 0 [deg]i.s. tslew 1.855 0 0 0 0 [s]i.s. tcal 4 0 0 0 0 [s]i.s. tcycle 300 300 300 300 300 [s]i.s. ttarget 292.2 300 300 300 300 [s]i.s. toverhead 7.711 0 0 0 0 [s]i.s. phase 3.620 0 0 0 0 [deg]i.s. atmophase 3.359 0 0 0 0 [deg]i.s. gainerror 1.348 0 0 0 0 [deg]same cal 0.377 0 0 0 0

Table 5: Median fast switching results for 140, 90, 70, and 35 GHz, giving the best conditions tothe high frequencies. Frequencies 90 GHz and lower use the target frequency for the calibrationfrequency. “t.s.” quantities are for an individual target sequence, “i.s.” quantities are for theinstrumental sequence, and “all” quantities refer to products of “t.s” and “i.s.” quantities.

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targetfreq 875 937 [GHz]calfreq 90 90 [GHz]el 60 60 [deg]tau225 0.029 0.029rms300 0.927 0.927 [deg @ 11.2GHz]all phase 35.25 37.20 [deg]phase drift 9.672 11.11 [deg]all eff0 0.707 0.682all eff1 0.854 0.843all eff2 0.827 0.809i.s. eff0 0.962 0.953i.s. eff1 0.973 0.967i.s. eff2 0.989 0.985t.s. eff0 0.735 0.716t.s. eff1 0.881 0.866t.s. eff2 0.837 0.823t.s. calflux 0.128 0.128 [Jy]t.s. caldist 1.797 1.798 [deg]t.s. tslew 1.663 1.664 [s]t.s. tcal 0.440 0.452 [s]t.s. tcycle 17.93 16.96 [s]t.s. Ncycles 15.85 16.56t.s. ttarget 13.70 12.89 [s]t.s. toverhead 4.090 4.122 [s]t.s. phase 34.15 35.74 [deg]t.s. atmophase 33.78 35.38 [deg]t.s. gainerror 7.590 7.942 [deg]i.s. calflux 1.087 1.298 [Jy]i.s. tarflux 0.312 0.362 [Jy]i.s. caldist 10.61 12.78 [deg]i.s. tslew 5.300 6.316 [s]i.s. tcal 4 4 [s]i.s. tcycle 300 300 [s]i.s. ttarget 284.4 280.9 [s]i.s. toverhead 15.59 19.01 [s]i.s. phase 8.285 9.867 [deg]i.s. atmophase 6.359 6.810 [deg]i.s. gainerror 5.311 7.071 [deg]same cal 0.052 0.035

Table 6: Median fast switching results for 5th percentile conditions, which will be used to helpset ALMA requirements.

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Figure 6: Histogram of the overall efficiency for the simulations reported on in Table 6 at875 GHz.

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7000 m2 collecting area, the telescope is much more sensitive. The shortest part of the fastswitching cycle is the time spent on the calibrator, so improving the telescope’s sensitivity doeslittle to improve fast switching with calibration observations at 90 GHz. However, it does alot to improve fast switching with the calibration observations being performed at the targetfrequency.

7.1 Requirements on the Instrumental Stability

We suggest that the instrumental stability requirements can be derived from the results inTable 6, which have been run for opacity and phase conditions corresponding to the best 5%of Chajnantor atmospheric conditions. We consider two different sorts of requirements: rmsinstrumental delay jitter on timescales of a few tens of seconds (ie, the time scale of a singlefast switching cycle), and the instrumental delay drift on time scales of about 300 s.

Our suggestion is that the instrumental phase fluctuations be equal to the atmosphericfluctuations in the 5th percentile atmospheric conditions. There is some ambiguity about whatthe 5th percentile conditions mean (we could take any pair of opacity and stability that lieson the 5th percentile contours of the joint probability distribution of D’Addario and Holdaway,2003), so we have taken those conditions which maximizes the array sensitivity at 875 and937 GHz, considering both sensitivity losses due to phase fluctuations and opacity. This turnsout to be close to 225 GHz opacity of 0.0296 and 11.2 GHz site testing interferometer phase of0.927 deg for 875 GHz.

For the delay jitter, the relevant parameter here is the “all phase” number in the Tablesabove. To convert from phase φ, in degrees, to delay τ (sorry for the multiple use of Greekletters again)

τ = φ/360 deg/ν.

This gives 112 fs for the 875 GHz case and 110 fs for the 937 GHz case. Hence, we suggestthat a baseline-based electronic and structural delay jitter specification for time scales up to afew tens of seconds of 111 fs is acceptable. Antenna-based would be 78 fs assuming that thephase fluctuations are independent among antennas.

Another number in the tables above is “phase drift”. Part of this number is from thefairly small phase residuals from the instrumental sequence, which get frozen in over the 300 sfull cycle; this is referred to as “i.s. phase” in the tables. There is also a contribution fromthe average phase jitter on each target sequence, though that averages down as 1/

√Ncycles.

However, the phase jitter is calculated using γ0, and for this determination, we need to use thesmaller γ1, which is appropriate for calculating the phase error. Hence, the phase drift overthe entire 300 s calibration cycle can be calculated as

drift =

√√√√((

γ1

γ0t.s.atmophase

)2

+ t.s.gainnoise2

)

/Ncycles + i.s.phase2 (8)

Regarding the phase drifts, we find that the 5th percentile visibility error due to residualatmospheric delay variation is 31 fs at 875 GHz and 33 fs at 937 GHz, so we’ll average thatto 32 fs. The instrumental error, is a combination of of drifts at the target frequency and thecalibration frequency, should be made smaller than this. A part of the drifts will be commonto both calibration and target bands (ie, antenna deformations or temperature-induced delaychanges), and they will not contribute to the scaled phase difference, except to the extent that

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the atmosphere at the target frequency is dispersive and a factor other than νtarget/νcal is usedto scale the calibrator phase to the target frequency. Because the determination of the fractionof the delay drift which will be common to the two frequency bands is complicated and outsidethe scope of this work, we will not make a suggested delay drift specification, but rather assertthat the drift in delay for the scaled frequency difference Φ̂i be 32 fs or better.

7.2 Impact of Instrumental Fluctuations

While the instrumental phase fluctuations of 111 fs will dominate the phase stability during onlyonly the best 5% of observing conditions, these fluctuations will impact all observations, thoughlow frequency observations are affected only marginally. Figure 7 shows how the instrumentalfluctuations would degrade observations through additional decorrelation across the ten ALMAbands, parsing up the atmospheric conditions the way we have in the previous section. Thisfigure is a bit bumpy because of odd variations in both the band sensitivity and the atmosphericconditions that are left for each lower frequency band. We define the integrated fractionalsensitivity loss to be due to additional decorrelation by the instrumental phase errors:

Nν∑

i

fi(1 − e−(2πτνi)2/2),

where the sum is over the frequency bands, fi is the normalized fraction of time we observein each band (ie, not counting time which is too bad to observe at any frequency), and τ isthe instrumental delay specification (ie, 111 fs). This sum ends up being 0.037, so the totalefficiency loss of ALMA due to this instrumental fluctuation specification, integrated over allbands, is under 4%. This is not an overly large burden, though the highest frequencies willcarry most of that burden.

8 References

D‘Addario, L., 2003, “Phase Calibration With Fast Switching: Implications for InstrumentalPhase Stability”, LAMA Memo 802.D‘Addario, L., and M.A. Holdaway, 2003, “Joint Distribution of Atmospheric Transparencyand Phase Fluctuations at Chajnantor”, LAMA Memo 801.

Butler, Bryan, 2003, “Distance to Possible Calibration Sources as a Function of Frequency forALMA”, ALMA Memo 478.

Condon, 1984, “Cosmological Evolution of Radio Sources”, ApJ, 287, 461.

Delgado, Guillermo et al., 2001, “Phase Cross-Correction of a 11.2 GHz Interferometer and183 GHz Water Line Radiometers at Chajnantor”, ALMA Memo 361.

Delgado, Guillermo et al., 2003, “Some Error Sources For The PWV And Path Delay Esti-mated From 183 GHZ Radiometric Measurements At Chajnantor”, ALMA Memo 451.

Feigelson, E.D., and Nelson, P.I., 1985, ApJ, 293, 192.

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Figure 7: The affect of 111 fs instrumental delay jitter on the efficiency due to decorrelation only(eff2) and the total efficiency (eff0). The values of eff2 and eff0 are taken straight from Tables 3-5 and show the effect of the residual atmosphere and the inefficiencies of the calibration process,but no instrumental delay errors. The curves labeled “+ inst” indicate how the efficiencies aredegraded when adding the 111 fs delay jitter.

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Hills, R., and H. Gibson, 2001, “Design and Development of 183 GHz Water Vapour Radiome-ters”, ALMA Memo 352.

Holdaway, M.A., F.N. Owen, and M.P. Rupen, 1994, “ Source Counts at 90 GHz”, ALMAMemo 123.

Holdaway, M.A., Simon J.E. Radford, F.N. Owen, and Scott M. Foster, 1995. “ Data Process-ing for Site Test Interferometers”, ALMA Memo 129.

Holdaway, M.A., 1996, “ Calibration of Sub-millimeter Observations with SiO Masers: Doesthe MMA Need Dual Band Capability?”, ALMA Memo 148.

Holdaway, M.A., and Mangum, Jeff, 2001, “Relative Pointing Sensitivity at 30 and 90 GHz forthe ALMA Test Interferometer”, ALMA Memo 373.

Holdaway, M.A., 2001, “Fast Switching Phase Correction Revisited for 64 12 m Antennas”,ALMA Memo 403.

Holdaway, M.A., and Pardo, J.R., 2001, “Atmospheric Dispersion and Fast Switching PhaseCalibration”, ALMA Memo 404.

Holdaway, M.A., 2003, “Effects of Atmospheric Emission Fluctuations and Gain Fluctuationson Continuum Total Power Observations with ALMA”, to appear as a LAMA Memo.

Holdaway, M.A., 2003b, unpublished simulations of decorrelation and its correction.

Patnaik et al, 1993, MNRAS 261, 435.

Radford, S.J.E., G. Reiland, and B. Shillue, Site Test Interferometer, PASP, 108, 441-445, 1996.

A Defining Terms

We define the following terms in reference to Figure 1. Most of these terms will appear in thesimulation results section in Tables 3 through 6.

• target sequence tcal is the time integrating on the weaker target sequence calibrator.

• target sequence tslew is the time it takes to slew from the target source to the targetsequence calibrator source, or twice the frequency change time if it is longer than theslew time.

• target sequence ttarget is the time spent on the target source at the target frequencyin a single target sequence cycle.

• target sequence toverhead includes both target sequence tcal and 2 * tslew, there andback, between the target source and the target sequence calibrator source.

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• target sequence tcycle is the target sequence ttarget plus the target sequence tover-head, the time for a single complete cycle.

• target sequence duty cycle is the target sequence time on source divided by the targetsequence cycle time.

• target sequence eff1 is the efficiency due only to time lost from the target source duringone target sequence cycle, and is given by the square root of the duty cycle.

• target sequence effective baseline is defined as vt/s + d, where v is the atmosphericvelocity and t is the cycle time, and d is the distance between lines of site to the sourceand the calibrator.

• target sequence atmophase σaφ is the atmospheric component of the target sequence

residual phase error at the target frequency, and is approximately equal to the phasestructure function evaluated at the target sequence effective baseline.

• target sequence gainerror is the error made in the gain phases, at the target frequency,due to finite SNR on the target sequence calibrator.

• target sequence phase σφ is the total phase error, rms, made by the fast target sequencecycles, and is equal to the sum in quadrature of the atmophase and the gainerror.

• target sequence eff2 is the efficiency of the calibration process due to the target se-quence phase errors, and is just the coherence given by

eff2 = e−σ2φ/2,

with σφ being the target sequence phase (due to atmosphere and gain errors) in radiansof phase at the target frequency.

• target sequence eff0 is the product of target sequence eff1 and eff2. As the targetsequence overhead time is determined by antenna capabilities and our need for accuratephase solutions, lengthening the target sequence cycle time will increase eff1, but willresult in larger residual phase errors and lower eff2. As the cycle time decreases, theopposite trends are in play. For very short cycle times, the overall target sequenceefficiency eff0 is low as eff1 tends toward zero. For very long cycle times, eff0 is low as eff2tends toward zero. In between, there is an optimal cycle time, which will depend upon theatmospheric phase fluctuations and all. ALMA should be programmed to dynamicallydetermine the target sequence cycle time which optimizes the target sequence efficiencyeff0.

• target sequence calibrator flux and calibrator distance are the flux of the cali-brator at the cal frequency, and the angular distance from the target source to the fastswitching calibrator.

• instrumental sequence tcal refers to the total time, in 1 s sub-integrations, spent onthe brighter instrumental sequence calibrator .

• instrumental sequence tslew is the time spent slewing from the target sequence phasecalibrator to the instrumental sequence calibrator source.

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• instrumental sequence ttarget is the time spent performing target sequence cycles(rather than instrumental sequence calibration).

• instrumental sequence toverhead is the instrumental sequence tcal plus twice theinstrumental sequence tslew.

• instrumental sequence cycle time (major cycle time) is the instrumental sequencettarget plus the instrumental sequence toverhead and is defined to be 300 s, the supposedinstrumental stability time scale, in this work.

• instrumental sequence duty cycle is the instrumental sequence time on source dividedby the instrumental sequence cycle time.

• instrumental sequence eff1 is the efficiency due only to time lost to the instrumentalsequence sequence or toverhead, and is the square root of the instrumental sequence dutycycle.

• scaled instrumental phase difference Φi will be the instrumental phase at the targetfrequency minus the instrumental phase at the cal frequency scaled by the ratio of thefrequencies:

Φi = φi(νtar) − φi(νcal) · νtar/νcal.

• instrumental sequence atmophase is the atmospheric contribution corrupting thescaled instrumental phase difference, and ends up being the root structure function eval-uated at about 2 s.

• instrumental sequence gainerror is the error in the scaled instrumental phase differ-ence Φi due to the influence of thermal noise on the gain phase determinations.

• instrumental sequence phase is the overall phase error in the determination of thescaled instrumental phase difference Φi, and is equal to the sum in quadrature of theinstrumental sequence gain error and the instrumental sequence atmophase.

• instrumental sequence eff2 is the efficiency of the instrumental sequence calibrationprocess due to instrumental sequence phase errors, and is equal to the coherence due tothe instrumental sequence phase errors.

• instrumental sequence eff0 is the product of instrumental sequence eff1 and eff2, andis a measure of the overall efficiency of the instrumental sequence calibration.

• instrumental sequence cal flux is the flux of the instrumental sequence calibrator atthe calibration frequency.

• instrumental sequence target flux is the flux of the instrumental sequence calibratorat the target frequency.

• all eff1 is the efficiency due to time lost from the target source by both target sequenceand instrumental sequence calibration.

• all eff2 is the efficiency due to residual phase errors, both atmospheric and noise-like,from both target sequence and instrumental sequence calibration.

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• all eff0 is the overall efficiency of the target sequence and instrumental sequence cali-bration processes, and is equal to all eff1 times all eff2. We seek a target sequence andinstrumental sequence strategy which optimize the quantity all eff0.

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